Exploring Monte Carlo Methods
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1 Exploring Monte Carlo Methods William L Dunn J. Kenneth Shultis AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO ELSEVIER Academic Press Is an imprint of Elsevier
2 Preface xv 1 Introduction What Is Monte Carlo? A Brief History of Monte Carlo Monte Carlo as Quadrature Monte Carlo as Simulation Preview of Things to Come IS 1.6 Summary 16 2 The Basis of Monte Carlo Single Continuous Random Variables Probability Density Function Cumulative Distribution Function Some Example Distributions Population Mean, Variance, and Standard Deviation Sample Mean, Variance, and Standard Deviation Discrete Random Variables Multiple Random Variables Two Random Variables More Than Two Random Variables Sums of Random Variables The Law of Large Numbers The Central Limit Theorem Monte Carlo Quadrature Monte Carlo Simulation Summary 44 3 Pseudorandom Number Generators Linear Congruential Generators Structure of the Generated Random Numbers Characteristics of Good Random Number Generators Tests for Congruential Generators Spectral Test Number of HyperpJanes Distance between Points Other Tests 53
3 viii Contents 3.5 Practical Multiplicative Congruential Generators Generators with m = 2" Prime Modulus Generators A Minimal Standard Congruential Generator Coding the Minimal Standard Deficiencies of the Minimal Standard Generator Optimum Multipliers for Prime Modulus Generators Shuffling a Generator's Output Skipping Ahead Combining Generators Bit Mixing The Wichmann-Hill Generator The L'Ecuyer Generator Other Random Number Generators Multiple Recursive Generators Lagged Fibonacci Generators Add-with-Carry Generators Inversive Congruential Generators Nonlinear Congruential Generators Summary 65 4 Sampling, Scoring, and Precision Sampling Inverse CDF Method for Continuous Variables Inverse CDF Method for Discrete Variables Rejection Method Composition Method Rectangle-Wedge-Tail Decomposition Method Sampling from a Nearly Linear PDF Composition-Rejection Method Ratio of Uniforms Method Sampling from a Joint Distribution Sampling from Specific Distributions Scoring Statistical Tests to Assess Results Scoring for "Successes-Over-Trials" Simulation Use of Weights in Scoring Scoring for Multidimensional Integrals Accuracy and Precision Factors Affecting Accuracy Factors Affecting Precision Summary 93 5 Variance Reduction Techniques Use of Transformations 100
4 ix 5.2 Importance Sampling Application to Monte Carlo Integration Systematic Sampling Comparison to Straightforward Sampling Systematic Sampling to Evaluate an Integral Systematic Sampling as Importance Sampling Stratified Sampling Comparison to Straightforward Sampling Importance Sampling Versus Stratified Sampling Correlated Sampling Correlated Sampling With One Known Expected Value Antithetic Variates Partition of the Integration Volume Reduction of Dimensionality Russian Roulette and Splitting Application to Monte Carlo Simulation Combinations of Different Variance Reduction Techniques Biased Estimators Improved Monte Carlo Integration Schemes Weighted Monte Carlo Integration Monte Carlo Integration with Quasirandom Numbers Summary Markov Chain Monte Carlo Markov Chains to the Rescue Ergodic Markov Chains The Metropolis-Hastings Algorithm The Myth of Burn-in The Proposal Distribution Multidimensional Sampling The Gibbs Sampler Brief Review of Probability Concepts Bayes Theorem Inference and Decision Applications Implementing MCMC with Data Summary Inverse Monte Carlo Formulation of the Inverse Problem Integral Formulation Practical Formulation Optimization Formulation Monte Carlo Approaches to Solving Inverse Problems Inverse Monte Carlo by Iteration Symbolic Monte Carlo Uncertainties in Retrieved Values 179
5 X Contents The PDF Is Fully Known The PDF Is Unknown Unknown Parameter in Domain of x Inverse Monte Carlo by Simulation A Simple Two-Dimensional World General Applications of IMC Summary Linear Operator Equations 8.1 Linear 203 Algebraic Equations Solution of Linear Equations by Random Walks Solution of the Adjoint Linear Equations by Random Walks Solution of Linear Equations by Finite Random Walks Linear Integral Equations Monte Carlo Solution of a Simple Integral Equation A More General Procedure for Integral Equations Linear Differential Equations Monte Carlo Solutions of Linear Differential Equations Continuous Monte Carlo for Laplace's Equation Generalization to Three Dimensions Continuous Monte Carlo for Poisson's Equation The Two-dimensional Helmholtz Equation The Three-dimensional Helmholtz Equation Eigenvalue Problems Summary The Fundamentals of Neutral Particle Transport Description of the Radiation Field Directions and Solid Angles Particle Density Flux Density Fluence Current Vector Radiation Interactions with the Medium Interaction Coefficient Macroscopic Cross Section Attenuation of Uncollided Radiation Average Travel Distance Before an Interaction Scattering Interaction Coefficients Microscopic Cross Sections Reaction Rate Density Transport Equation Integral Forms of the Transport Equation Adjoint Transport Equation Derivation of the Adjoint Transport Equation 262
6 xi Utility of the Adjoint Solution Summary Monte Carlo Simulation of Neutral Particle Transport Basic Approach for Monte Carlo Transport Simulations Geometry Combinatorial Geometry Sources Isotropic Sources Path-Length Estimation Travel Distance in Each Cell Convex versus Concave Cells Effect of Computer Precision Purely Absorbing Media Type of Collision Scattering Interactions Photon Scattering from a Free Electron Neutron Scattering Time Dependence Particle Weights Scoring and Tallies Fluence Averaged Over a Surface Average Fluence in a Volume: Path-Length Estimator Average Fluence in a Volume: Reaction-Density Estimator Average Current Through a Surface Fluence at a Point: Next-Event Estimator Flow Through a Surface: Leakage Estimator An Example of One-Speed Particle Transport Monte Carlo Based on the Integral Transport Equation Integral Transport Equation Integral Equation Method as Simulation Variance Reduction and Nonanalog Methods Importance Sampling Truncation Methods Splitting and Russian Roulette Implicit Absorption Interaction Forcing Exponential Transformation Summary 304 A Some Common Probability Distributions 307 A.l Continuous Distributions 307 A. 1.1 Uniform Distribution 308 A. 1.2 Exponential Distribution 310 A. 1.3 Gamma Distribution 312
7 xii Contents A. 1.4 Beta Distribution 314 A. 1.5 Weibull Distribution 316 A. 1.6 Normal Distribution 317 A. 1.7 Lognormal Distribution 319 A. 1.8 Cauchy Distribution 319 A. 1.9 Chi-Squared Distribution 321 A.1.10 Student's t Distribution 322 A Pareto Distribution 323 A.2 Discrete Distributions 324 A.2.1 Bernoulli Distribution 325 A.2.2 Binomial Distribution 326 A.2.3 Geometric Distribution 329 A.2.4 Negative Binomial Distribution 329 A.2.5 Poisson Distribution 330 A. 3 Joint Distributions 333 A.3.1 Multivariate Normal Distribution 333 A. 3.2 Multinomial Distribution 334 B The Weak and Strong Laws of Large Numbers 337 B. 1 The Weak Law of Large Numbers 337 B. 2 The Strong Law of Large Numbers 339 B. 2.1 Difference Between the Weak and Strong Laws 339 B. 2.2 Other Subtleties 340 C Central Limit Theorem 341 C. l Moment Generating Functions 341 C. l.l Central Moments 342 C. 1.2 Some Properties of the MGF 342 C. 1.3 Uniqueness of the MGF 344 C. 2 Central Limit Theorem 344 D Some Popular Monte Carlo Codes for Particle Transport 347 D. l COG 347 D. l.l Current Version 347 D.1.2 Principal Authors 347 D.1.3 Operating Systems 347 D.1.4 Availability 347 D. 1.5 Applications 348 D.l.6 Significant Features 348 D.2 EGSnrc 349 D.2.1 Acronym and Current Version 349 D.2.2 Principal Authors 349 D.2.3 Operating Systems 349 D.2.4 Availability 349 D.2.5 Applications 349 D.2.6 Significant Features 350
8 xiii D.3 GEANT4 351 D.3.1 Current Version 351 D.3.2 Principal Author 351 D.3.3 Operating Systems 351 D.3.4 Availability 351 D.3.5 Applications 352 D.3.6 Significant Features 352 D.4 MCNP and MCNPX 353 D.4.1 Current Version and Availability 354 D.4.2 Principal Authors 354 D.4.3 Operating Systems 354 D.4.4 Applications 354 D.4.5 Significant Features 355 D.4.6 Additional Information 356 D.5 MCSHAPE 356 D.5.1 Acronyms 356 D.5.2 Current Version and Availability 356 D.5.3 Principal Authors 357 D.5.4 Operating Systems 357 D.5.5 Applications 357 D.5.6 Significant Features 358 D.6 PENELOPE 359 D.6.1 Acronyms and Current Version 359 D.6.2 Principal Authors 359 D.6.3 Operating Systems 359 D.6.4 Availability 359 D.6.5 Applications 359 D.6.6 Significant Features 361 D.7 SCALE 361 D.7.1 Acronyms 361 D.7.2 Current Version 362 D.7.3 Principal Author 362 D.7.4 Operating Systems 362 D.7.5 Availability 362 D.7.6 Applications 362 D.7.7 Significant Features 363 D.8 SRIM 364 D.8.1 Acronyms 364 D.8.2 Principal Authors 364 D.8.3 Operating Systems 364 D.8.4 Availability 364 D.8.5 Applications 364 D.8.6 Significant Features 365 D.9 TRIPOLI 365 D.9.1 Acronym and Current Version 365
9 D.9.2 Principal Authors 365 D.9.3 Operating Systems 366 D.9.4 Availability 366 D.9.5 Applications 366 D.9.6 Significant Features 366 E Minimal Standard Pseudorandom Number Generator 373 E.l FORTRAN E.2 FORTRAN E.3 Pascal 374 E.4 C and C E.5 Programming Considerations 375 Index 377
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